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Local and global optima: Sheaves and polynomial approximations

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  • Tohmé, Fernando

Abstract

This paper develops a sheaf-theoretic framework to reconstruct an economic agent’s global preference optima from solutions to local decision problems. We formalize how local utility maximizations, represented as a category of problems, can be consistently glued to approximate the global utility function using a contravariant functor and sheaf properties. When only a limited number of local solutions are available, we propose polynomial approximations to construct a global utility function, ensuring minimal complexity via a Gröbner basis approach. This contribution presents a categorical characterization of global-local relationships, conditions for unique global solutions under concave utilities, and a polynomial-based method to approximate optima in general cases.

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  • Tohmé, Fernando, 2026. "Local and global optima: Sheaves and polynomial approximations," Applied Mathematics and Computation, Elsevier, vol. 517(C).
    • Handle: RePEc:eee:apmaco:v:517:y:2026:i:c:s0096300325006290
    DOI: 10.1016/j.amc.2025.129904
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